How would one independently measure the distance to the outer planets?

Question

I know how the ancients measured the Earth, Moon, Sun and (after Copernicus) Venus and Mercury. But how would a single observer on Earth be able to measure the distance to the outer worlds? (I could only get the interior ones using observations at the greatest elongations, and this doesnt help when a world doenst have those to measure).

Answer 1

Newton came up with the math that lets you determine the equations of motion for any object using three observations.  You have to know where you are on Earth, which way you are looking, and what time it is.  You can improve precision by doing more observations.  The math is tricky, but JPL has free software that you can use.

Before that, you could assume that the outer planets didn't move much compared to the Earth and use parallax.  You make an observation, wait a bit, and make another.  The Earth has moved in the mean time, and you see what angle the planet is at between observations.  It's not very good because you are making an assumption that is wrong. But this is kind of how Newton's math works too, only it doesn't quite assume that the planet didn't move.

Uranus and Neptune were discovered after Newton's time.  In fact, Newton's math was used to find Neptune.

Answer 2

The easiest way is to measure the phase of the lighted disk at solar quadrature. That gives you the distance in AU. It requires a telescope, though.

At solar quadrature, the Earth, Sun and planet form a right-triangle with the Earth at the right angle (i.e., the Sun and planet are 90° apart as viewed from Earth). The phase angle of the planet's lighted disk tells you the angle between the Sun and Earth as seen from the planet. Then you simply solve the triangle using normal trig.

Answer 3

parallax, kepler's 3rd law.